Symmetries of algebras captured by actions of weak Hopf algebras

Abstract

In this paper, we present a generalization of well-established results regarding symmetries of $\mathbb{k}$-algebras, where $\mathbb{k}$ is a field. Traditionally, for a $\mathbb{k}$-algebra $A$, the group $\mathbb{k}$-algebra automorphisms of $A$ captures the symmetries of $A$ via group actions. Similarly, the Lie algebra of derivations of $A$ captures the symmetries of $A$ via Lie algebra actions. In this paper, given a category $\mathcal{C}$ whose objects possess $\mathbb{k}$-linear monoidal categories of modules, we introduce an object ${\mathrm{Sym}}_{\mathcal{C}}(A)$ that captures the symmetries of $A$ via actions of objects in $\mathcal{C}$. Our study encompasses various categories whose objects include groupoids, Lie algebroids, and more generally, cocommutative weak Hopf algebras. Notably, we demonstrate that for a positively graded non-connected $\mathbb{k}$-algebra $A$, some of its symmetries are naturally captured within the weak Hopf framework.

v1: 34 pages; comments welcome. v2: 45 pages; name changed, sections reorganized, new examples. v3: 47 pages; minor changes and clarifications added; final version to appear in Algebras and Representation Theory

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• On anchored Lie algebras and the Connes-Moscovici's bialgebroid construction
• Algebraic structures in comodule categories over weak bialgebras